
Hence, KL divergence converges exponentially fast along the Langevin
dynamics. Log-Sobolev inequality can be thought as a relaxation of
logconcavity in continuous time. LSI was originally initiated by [1]}
for the scenario of Gaussian measure, characterizes concentration
of measure and sub-Gaussian tail property, to name a few. [2]}
broadened it to strongly log-concave measure, where \(\pi \)  enjoys
LSI with constant \(\gamma \)  whenever \(-\log \pi \)  is \(\gamma \) -strongly
convex. However, LSI is more general than strongly log-concave condition
since it is preserved under bounded perturbation [3]},
Lipschitz mapping, tensorization, among others. Therefore, we will
study the KL convergence under log-Sobolev inequality.
